A device of this nature is known from U.S. Pat. No. 2,344,127. This device is used to handle blind rivets which are suspended one behind the other in a rail and in the rail are forced towards the end of the rail by spring stress. At the end of the rail, there is a mechanism which closes off the opening of the rail which is present in that area and allows only a single blind rivet to be delivered each time. This delivery is carried out by means of a lever which is arranged on a setting tool and is mounted on the setting tool in such a manner that it can pivot about an axis. To deliver a blind rivet which is suspended in front of the opening in the rail, the lever is pivoted towards this opening, during which movement the lever is able to pick up a single blind rivet and pivot it into a position in which the blind rivet is inserted in the setting tool. The rail is arranged on the side of the setting tool and thus forms a single unit with the setting tool. The mechanism which is required to deliver a blind rivet while simultaneously unlocking the rail is of complicated design and is arranged on the setting tool in such a manner that it can be accessed freely from outside, so that if the tool is not handled carefully the delivery mechanism can easily become damaged. Moreover, this device is a relatively complicated structure.